Optimal. Leaf size=144 \[ -\frac {\sqrt {f+g x} \left (a e^2+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac {\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac {2 \left (a g^2+c f^2\right )}{g \sqrt {f+g x} (e f-d g)^2} \]
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Rubi [A] time = 0.27, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {898, 1259, 453, 208} \begin {gather*} -\frac {\sqrt {f+g x} \left (a e^2+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac {\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac {2 \left (a g^2+c f^2\right )}{g \sqrt {f+g x} (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 453
Rule 898
Rule 1259
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}-\frac {g^3 \operatorname {Subst}\left (\int \frac {\frac {2 e^2 (e f-d g) \left (c f^2+a g^2\right )}{g^5}+\frac {e \left (a e^2 g^2-c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )} \, dx,x,\sqrt {f+g x}\right )}{e^2 (e f-d g)^2}\\ &=-\frac {2 \left (c f^2+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}-\frac {\left (3 a e^2 g+c d (4 e f-d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e g (e f-d g)^2}\\ &=-\frac {2 \left (c f^2+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}+\frac {\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 118, normalized size = 0.82 \begin {gather*} -\frac {2 \left (g^2 \left (a e^2+c d^2\right ) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )+2 c d g (e f-d g) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )+c (e f-d g)^2\right )}{e^2 g \sqrt {f+g x} (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.57, size = 210, normalized size = 1.46 \begin {gather*} \frac {2 a d e g^3+3 a e^2 g^2 (f+g x)-2 a e^2 f g^2+c d^2 g^2 (f+g x)+2 c d e f^2 g-2 c e^2 f^3+2 c e^2 f^2 (f+g x)}{e g \sqrt {f+g x} (e f-d g)^2 (-d g-e (f+g x)+e f)}+\frac {\left (3 a e^2 g-c d^2 g+4 c d e f\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x} \sqrt {d g-e f}}{e f-d g}\right )}{e^{3/2} (e f-d g)^2 \sqrt {d g-e f}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 906, normalized size = 6.29 \begin {gather*} \left [\frac {{\left (4 \, c d^{2} e f^{2} g - {\left (c d^{3} - 3 \, a d e^{2}\right )} f g^{2} + {\left (4 \, c d e^{2} f g^{2} - {\left (c d^{2} e - 3 \, a e^{3}\right )} g^{3}\right )} x^{2} + {\left (4 \, c d e^{2} f^{2} g + 3 \, {\left (c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} - 3 \, a d e^{2}\right )} g^{3}\right )} x\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c d e^{3} f^{3} - 2 \, a d^{2} e^{2} g^{3} - {\left (c d^{2} e^{2} - a e^{4}\right )} f^{2} g - {\left (c d^{3} e - a d e^{3}\right )} f g^{2} + {\left (2 \, c e^{4} f^{3} - 2 \, c d e^{3} f^{2} g + {\left (c d^{2} e^{2} + 3 \, a e^{4}\right )} f g^{2} - {\left (c d^{3} e + 3 \, a d e^{3}\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{2 \, {\left (d e^{5} f^{4} g - 3 \, d^{2} e^{4} f^{3} g^{2} + 3 \, d^{3} e^{3} f^{2} g^{3} - d^{4} e^{2} f g^{4} + {\left (e^{6} f^{3} g^{2} - 3 \, d e^{5} f^{2} g^{3} + 3 \, d^{2} e^{4} f g^{4} - d^{3} e^{3} g^{5}\right )} x^{2} + {\left (e^{6} f^{4} g - 2 \, d e^{5} f^{3} g^{2} + 2 \, d^{3} e^{3} f g^{4} - d^{4} e^{2} g^{5}\right )} x\right )}}, -\frac {{\left (4 \, c d^{2} e f^{2} g - {\left (c d^{3} - 3 \, a d e^{2}\right )} f g^{2} + {\left (4 \, c d e^{2} f g^{2} - {\left (c d^{2} e - 3 \, a e^{3}\right )} g^{3}\right )} x^{2} + {\left (4 \, c d e^{2} f^{2} g + 3 \, {\left (c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} - 3 \, a d e^{2}\right )} g^{3}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, c d e^{3} f^{3} - 2 \, a d^{2} e^{2} g^{3} - {\left (c d^{2} e^{2} - a e^{4}\right )} f^{2} g - {\left (c d^{3} e - a d e^{3}\right )} f g^{2} + {\left (2 \, c e^{4} f^{3} - 2 \, c d e^{3} f^{2} g + {\left (c d^{2} e^{2} + 3 \, a e^{4}\right )} f g^{2} - {\left (c d^{3} e + 3 \, a d e^{3}\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{d e^{5} f^{4} g - 3 \, d^{2} e^{4} f^{3} g^{2} + 3 \, d^{3} e^{3} f^{2} g^{3} - d^{4} e^{2} f g^{4} + {\left (e^{6} f^{3} g^{2} - 3 \, d e^{5} f^{2} g^{3} + 3 \, d^{2} e^{4} f g^{4} - d^{3} e^{3} g^{5}\right )} x^{2} + {\left (e^{6} f^{4} g - 2 \, d e^{5} f^{3} g^{2} + 2 \, d^{3} e^{3} f g^{4} - d^{4} e^{2} g^{5}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 225, normalized size = 1.56 \begin {gather*} \frac {{\left (c d^{2} g - 4 \, c d f e - 3 \, a g e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{{\left (d^{2} g^{2} e - 2 \, d f g e^{2} + f^{2} e^{3}\right )} \sqrt {d g e - f e^{2}}} - \frac {{\left (g x + f\right )} c d^{2} g^{2} + 2 \, c d f^{2} g e + 2 \, a d g^{3} e + 2 \, {\left (g x + f\right )} c f^{2} e^{2} - 2 \, c f^{3} e^{2} + 3 \, {\left (g x + f\right )} a g^{2} e^{2} - 2 \, a f g^{2} e^{2}}{{\left (d^{2} g^{3} e - 2 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )} {\left (\sqrt {g x + f} d g + {\left (g x + f\right )}^{\frac {3}{2}} e - \sqrt {g x + f} f e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 269, normalized size = 1.87 \begin {gather*} -\frac {3 a e g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) e}}+\frac {c \,d^{2} g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) e}\, e}-\frac {4 c d f \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) e}}-\frac {\sqrt {g x +f}\, a e g}{\left (d g -e f \right )^{2} \left (e g x +d g \right )}-\frac {\sqrt {g x +f}\, c \,d^{2} g}{\left (d g -e f \right )^{2} \left (e g x +d g \right ) e}-\frac {2 a g}{\left (d g -e f \right )^{2} \sqrt {g x +f}}-\frac {2 c \,f^{2}}{\left (d g -e f \right )^{2} \sqrt {g x +f}\, g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.29, size = 187, normalized size = 1.30 \begin {gather*} -\frac {\frac {2\,\left (c\,f^2+a\,g^2\right )}{d\,g-e\,f}+\frac {\left (f+g\,x\right )\,\left (c\,d^2\,g^2+2\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{e\,{\left (d\,g-e\,f\right )}^2}}{\sqrt {f+g\,x}\,\left (d\,g^2-e\,f\,g\right )+e\,g\,{\left (f+g\,x\right )}^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {f+g\,x}\,\left (d^2\,e\,g^2-2\,d\,e^2\,f\,g+e^3\,f^2\right )}{\sqrt {e}\,{\left (d\,g-e\,f\right )}^{5/2}}\right )\,\left (-c\,g\,d^2+4\,c\,f\,d\,e+3\,a\,g\,e^2\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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